Syllabus ( MATH 447 )
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Basic information
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| Course title: |
Tensor Analysis |
| Course code: |
MATH 447 |
| Lecturer: |
Prof. Dr. Oğul ESEN
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| ECTS credits: |
5 |
| GTU credits: |
3 (3+0+0) |
| Year, Semester: |
4, Fall |
| Level of course: |
First Cycle (Undergraduate) |
| Type of course: |
Elective
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| Language of instruction: |
English
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| Mode of delivery: |
Face to face
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| Pre- and co-requisites: |
Math 113 or Math 116 |
| Professional practice: |
No |
| Purpose of the course: |
To teach the concept of invariance and the use of tensors as a mathematical tool necessary for invariance, as well as to explain the similarities and differences with simpler concepts previously learned, such as scalars and vectors. |
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Learning outcomes
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Upon successful completion of this course, students will be able to:
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Build up a solid background of tensor calculus.
Contribution to Program Outcomes
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Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
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Describing, formulating, and analyzing real-life problems using mathematical and statistical techniques.
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Ability to work in interdisciplinary research teams effectively.
Method of assessment
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Written exam
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Use the basic concepts of applied mathematics
Contribution to Program Outcomes
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Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
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Describing, formulating, and analyzing real-life problems using mathematical and statistical techniques.
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Exhibiting professional and ethical responsibility.
Method of assessment
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Written exam
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The students learn the application of Tensors which are the most important tools for the concept of Invariance.
Contribution to Program Outcomes
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Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
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Communicating between mathematics and other disciplines, and building mathematical models for interdisciplinary problems.
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Describing, formulating, and analyzing real-life problems using mathematical and statistical techniques.
Method of assessment
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Written exam
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Contents
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| Week 1: |
The concept of invariance, physical transformations and the concept of invariant equations. |
| Week 2: |
Translational and rotational transformations. |
| Week 3: |
Lorentz transformations and the broken invariance of Newton equation under Lorentz transformations. |
| Week 4: |
Scalar invariants under these transformations. The infinitesimal arc length and its basis situation for tensorel calculations. |
| Week 5: |
Vectors and their invariance principles. 2nd order and higher order tensors. |
| Week 6: |
Maxwell equations and their invariance under Lorentz transformations. |
| Week 7: |
Relativistic electrodynamics and its formalism. |
| Week 8: |
Riemannian geometry and its principal tensors. Midterm Exam. |
| Week 9: |
Ricci and Riemann tensors. |
| Week 10: |
Einstein field equations and general relativity as a tensorel theory. |
| Week 11: |
Various examples to the solutions of Einstein field equations. |
| Week 12: |
Main solution techniques and the analysis of general relativity. |
| Week 13: |
Lagrange theory of fields and writing invariant action functional. The application of tensors to physical sciences. |
| Week 14: |
Invariant field theories and their difference from coordinate based theories. |
| Week 15*: |
- |
| Week 16*: |
Final Exam. |
| Textbooks and materials: |
General Relativity, Robert M. Wald, The University of Chicago Press, 1984
A Short Course in General Relativity, J. Foster, J. D. Nightingale, Springer-Verlag, 1995 |
| Recommended readings: |
Mathematical Methods for Physicists (Weber and Arfken)
Tensors, Differential Forms and Variational Principles (David Lovelock and Hanno Rund)
Tensor Analysis for Physicists (J. A. Schouten)
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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Assessment
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| Method of assessment |
Week number |
Weight (%) |
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| Mid-terms: |
8 |
40 |
| Other in-term studies: |
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0 |
| Project: |
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0 |
| Homework: |
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0 |
| Quiz: |
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0 |
| Final exam: |
16 |
60 |
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Total weight: |
(%) |
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Workload
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| Activity |
Duration (Hours per week) |
Total number of weeks |
Total hours in term |
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| Courses (Face-to-face teaching): |
3 |
14 |
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| Own studies outside class: |
3 |
14 |
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| Practice, Recitation: |
0 |
0 |
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| Homework: |
0 |
0 |
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| Term project: |
0 |
0 |
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| Term project presentation: |
0 |
0 |
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| Quiz: |
0 |
0 |
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| Own study for mid-term exam: |
12 |
1 |
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| Mid-term: |
3 |
1 |
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| Personal studies for final exam: |
20 |
1 |
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| Final exam: |
3 |
1 |
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Total workload: |
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Total ECTS credits: |
* |
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* ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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